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On-shell supersymmetry for massive multiplets

Rutger H. Boels, Christian Schwinn

TL;DR

This work develops a covariant on-shell framework for four-dimensional massive supermultiplets by extending the spinor-helicity formalism and introducing on-shell superspaces via coherent states. It derives massive SUSY transformations, formulates superamplitudes, and solves SUSY Ward identities, enabling explicit tree-level amplitudes and supersymmetric on-shell recursion for theories with massive particles. The paper applies the formalism to SQCD with massive matter, the Abelian Higgs model, the Coulomb branch of N=4, and Higgs–gluon effective couplings, illustrating three-point vertices and all-multiplicity MHV-type amplitudes. It also discusses off-shell elements like vector-boson currents and outlines how supersymmetric recursions handle residues at infinity, with implications for extensions to extended SUSY, string theory connections, and potential broken-SUSY contexts.

Abstract

The consequences of on-shell supersymmetry are studied for scattering amplitudes with massive particles in four dimensions. Using the massive version of the spinor helicity formalism the supersymmetry transformations relating products of on-shell states are derived directly from the on-shell supersymmetry algebra for any massive representation. Solutions to the resulting Ward identities can be constructed as functions on the on-shell superspaces that are obtained from the coherent state method. In simple cases it is shown that these superspaces allow one to construct explicitly supersymmetric scattering amplitudes. Supersymmetric on-shell recursion relations for tree-level superamplitudes with massive particles are introduced. As examples, simple supersymmetric amplitudes are constructed in SQCD, the Abelian Higgs model, the Coulomb branch of N=4 super Yang-Mills, QCD with an effective Higgs-gluon coupling and for massive vector boson currents.

On-shell supersymmetry for massive multiplets

TL;DR

This work develops a covariant on-shell framework for four-dimensional massive supermultiplets by extending the spinor-helicity formalism and introducing on-shell superspaces via coherent states. It derives massive SUSY transformations, formulates superamplitudes, and solves SUSY Ward identities, enabling explicit tree-level amplitudes and supersymmetric on-shell recursion for theories with massive particles. The paper applies the formalism to SQCD with massive matter, the Abelian Higgs model, the Coulomb branch of N=4, and Higgs–gluon effective couplings, illustrating three-point vertices and all-multiplicity MHV-type amplitudes. It also discusses off-shell elements like vector-boson currents and outlines how supersymmetric recursions handle residues at infinity, with implications for extensions to extended SUSY, string theory connections, and potential broken-SUSY contexts.

Abstract

The consequences of on-shell supersymmetry are studied for scattering amplitudes with massive particles in four dimensions. Using the massive version of the spinor helicity formalism the supersymmetry transformations relating products of on-shell states are derived directly from the on-shell supersymmetry algebra for any massive representation. Solutions to the resulting Ward identities can be constructed as functions on the on-shell superspaces that are obtained from the coherent state method. In simple cases it is shown that these superspaces allow one to construct explicitly supersymmetric scattering amplitudes. Supersymmetric on-shell recursion relations for tree-level superamplitudes with massive particles are introduced. As examples, simple supersymmetric amplitudes are constructed in SQCD, the Abelian Higgs model, the Coulomb branch of N=4 super Yang-Mills, QCD with an effective Higgs-gluon coupling and for massive vector boson currents.

Paper Structure

This paper contains 60 sections, 251 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Massive spinor helicity formalism
  3. Massive one-particle states with a fixed spin axis
  4. Polarization spinors of massive spin one-half particles
  5. Polarization vectors of massive spin one-bosons
  6. Special choice of frame
  7. Lorentz-invariance constraints on amplitudes
  8. Covariant representation theory of the $\mathcal{N}=1$, $D=4$ SUSY algebra
  9. Spin decomposition of the SUSY generators
  10. General massive representations in an arbitrary frame
  11. Massive quark multiplet
  12. Massive vector multiplet
  13. Extended supersymmetry and BPS multiplets
  14. On-shell superspaces for the masses
  15. Coherent states for massless $\mathcal{N}=1$ SUSY
  16. ...and 45 more sections

Figures (2)

  • Figure 1: Sketch of the states in the massive multiplet and the action of the supercharges.
  • Figure 2: Example topologies contributing to the six point amplitudes of massive vectors in the Abelian Higgs model.